Is there a way to avoid a Smooth modifier interfering with a bump-map?
I would suggest a different method to get the same results. Here's what's happening with what you have right now:

The bevel modifier is adding a small bevel on each edge of the cube, which adds some extra vertices. The array modifier duplicates this object along a set axes, and then the smooth modifier smooths out all of the geometry.

The smooth modifier doesn't create any new geometry. Instead, the relatively high strength you are using is smoothing the object so much that the originally small bevels are now much larger. Essentially, the vertices created by the bevel modifier are getting pushed towards the center of the cube.

This is all fine, but you're noticing an issue that this method causes.

Materials are applied to the object before certain modifiers. Thus, the texture is applied to the bevel created by the bevel modifier, but then that bevel is stretched (smoothed) by the smooth modifier, and thus you get the stretched texture.

Instead, remove the bevel modifier and the smooth modifier, and use a subdivision surface modifier set to simple. This will add some extra vertices that will allow the curve to bend it smoothly.